What is the integral from 0 to 10 of 8000e^{-0.005t} ?

The integral from 0 to 10 of 8000e^{-0.005t} is 78032.92079…

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integral from 0 to 10 of 8000e^{-0.005t}

\int_{0}^{10} 8000 e^{- 0.005 t} d t

78032.92079 \dots

Solution steps

\int_{0}^{10} 8000 e^{- 0.005 t} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8000 \cdot \int_{0}^{10} e^{- 0.005 t} d t

Apply u-substitution

= 8000 \cdot \int_{0}^{- 0.05} - \frac{1}{0.005} e^{u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 8000 (- \int_{- 0.05}^{0} - \frac{1}{0.005} e^{u} d u)

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8000 (- (- \frac{1}{0.005} \cdot \int_{- 0.05}^{0} e^{u} d u))

Simplify

= 8000 (- (- 200 \cdot \int_{- 0.05}^{0} e^{u} d u))

Use the common integral:

\int e^{u} d u = e^{u}

= 8000 (- (- 200 [e^{u}]_{- 0.05}^{0}))

Simplify

= 1600000 [e^{u}]_{- 0.05}^{0}

Compute the boundaries:

0.04877 \dots

= 1600000 \cdot 0.04877 \dots

Simplify

= 78032.92079 \dots

What is the integral from 0 to 10 of 8000e^{-0.005t} ?
The integral from 0 to 10 of 8000e^{-0.005t} is 78032.92079…